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  <title>数学-高等数学 0 第1讲 高数预备知识</title>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#1">第1讲 高数预备知识</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1-a_1-dd-neq-0">1. 等差数列（首项  <script type="math/tex"> a_1 </script> ，公差  <script type="math/tex"> d(d \neq 0) </script>  ）</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2-a_1-rr-neq-0">2. 等比数列（首项  <script type="math/tex"> a_1 </script> ，公比  <script type="math/tex"> r(r \neq 0) </script> ）</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3">3. 三角函数基本关系</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_1">1）倍角公式</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2">2）和差公式</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_1">3）积化和差公式</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4">4）和差化积公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_1">4. 因式子分解公式</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5">5. 常用不等式</a>
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  <h1 id="数学-高等数学 0 第1讲 高数预备知识" class="content-subhead">数学-高等数学 0 第1讲 高数预备知识</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h2 id="1">第1讲 高数预备知识</h2>
<h3 id="1-a_1-dd-neq-0">1. 等差数列（首项  <script type="math/tex"> a_1 </script> ，公差  <script type="math/tex"> d(d \neq 0) </script>  ）</h3>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1 + (n-1)d
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = \cfrac{n(a_1+a_n)}{2}
</script>
</p>
<h3 id="2-a_1-rr-neq-0">2. 等比数列（首项  <script type="math/tex"> a_1 </script> ，公比  <script type="math/tex"> r(r \neq 0) </script> ）</h3>
<p>通项公式</p>
<p>
<script type="math/tex; mode=display">
a_n = a_1r^{(n-1)}
</script>
</p>
<p>前  <script type="math/tex"> n </script>  项的和</p>
<p>
<script type="math/tex; mode=display">
S_n = 
\begin{cases}
na_1, & \text{r = 1} \\[2ex]
\cfrac{a_1(1-r^n)}{1-r}, & r \neq 1
\end{cases}
</script>
</p>
<p>常用  <script type="math/tex"> 1 + r + r^2 + \cdots + r^{n-1} = \cfrac{1 - r^n}{1 - r} </script>
</p>
<h3 id="3">3. 三角函数基本关系</h3>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学.assets/三角函数.jpg" alt="sin_cos" style="zoom:33%;" /></p>
<h4 id="1_1">1）倍角公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin2\alpha &= 2\sin\alpha *\cos\alpha \\[2ex]
\cos2\alpha &=\cos^2\alpha -\sin^2\alpha \\ 
\quad &= 1 - 2\sin^2\alpha \\ 
\quad &= 2\cos^2\alpha - 1 \\[2ex]
\tan2\alpha &= \cfrac{2 \tan\alpha}{1 - \tan^2\alpha}
\end{split}\end{equation}
</script>
</p>
<h4 id="2">2）和差公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin(\alpha \pm \beta) &=\sin\alpha\cos\beta \pm\cos\alpha\sin\beta \\
\sin(\alpha \pm \beta) &=\cos\alpha\cos\beta \mp\sin\alpha\sin\beta \\
\tan(\alpha \pm \beta) &= \cfrac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}
\end{split}\end{equation}
</script>
</p>
<h4 id="3_1">3）积化和差公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha\cos\beta &= \frac{1}{2}\bigg[\sin(\alpha + \beta) +\sin(\alpha - \beta)\bigg] \\
\cos\alpha\sin\beta &= \frac{1}{2}\bigg[\sin(\alpha + \beta) -\sin(\alpha - \beta)\bigg] \\
\sin\alpha\sin\beta &= \frac{1}{2}\bigg[\cos(\alpha + \beta) +\cos(\alpha - \beta)\bigg] \\
\cos\alpha\cos\beta &= \frac{1}{2}\bigg[\cos(\alpha - \beta) -\cos(\alpha + \beta)\bigg]
\end{split}\end{equation}
</script>
</p>
<h4 id="4">4）和差化积公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\sin\alpha +\sin\beta &= \quad 2\sin\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\sin\alpha -\sin\beta &= \quad 2\sin\frac{\alpha - \beta}{2}\cos\frac{\alpha + \beta}{2} \\
\cos\alpha +\cos\beta &= \quad 2\cos\frac{\alpha + \beta}{2}\cos\frac{\alpha - \beta}{2} \\
\cos\alpha -\cos\beta &= -\ 2\sin\frac{\alpha + \beta}{2}\sin\frac{\alpha - \beta}{2}
\end{split}\end{equation}
</script>
</p>
<h3 id="4_1">4. 因式子分解公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
(a + b)^n &= C_n^0a^n + C_n^1a^{n-1}b + ... + C_n^nb^n \\[2ex]
a^n - b^n &= (a - b)(a^n + a^{n-1}b + \cdots + ab^{n-1} + b^n) \\
\end{split}\end{equation}
</script>
</p>
<h3 id="5">5. 常用不等式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split} 
\bigg\vert\vert a \vert - \vert b \vert\bigg\vert &\le \vert a \pm b \vert \le \vert a \vert + \vert b \vert \\[4ex]
调和平均\le 几何平均 &\le 算数平均 \le 平方根平均 \\[2ex]
\cfrac{2}{\frac{1}{a}+\frac{1}{b}} \le \sqrt{ab} &\le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} \ (a,b \gt 0) \\[2ex]
\cfrac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} \le\sqrt[3]{abc} &\le \frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \ (a,b,c \gt 0) \\[4ex]
\sin x &\lt x \lt \tan x \ (0 \lt x \lt \frac{\pi}{2})
\end{split}\end{equation}
</script>
</p>
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